Topology Aware Neural Interpolation of Scalar Fields

2025-08-25

AI Summary

This work addresses topology-aware interpolation of time-varying scalar fields. Given sparse keyframes and their corresponding persistence diagrams, the proposed method learns a time-to-field mapping via a neural network, leveraging persistence diagrams of non-keyframes as topological constraints to reconstruct missing temporal samples. A novel topological loss is introduced to jointly optimize geometric fidelity and topological consistency, enabling end-to-end, real-time interpolation. The approach achieves state-of-the-art performance on standard 2D and 3D benchmark datasets, outperforming conventional interpolation methods in both reconstruction accuracy and topological fidelity. Quantitative and qualitative evaluations demonstrate improved preservation of critical topological features—such as connected components and holes—across time. By embedding persistent homology directly into the learning objective, the framework establishes a new paradigm for efficient, interpretable, and topology-preserving interpolation of time-varying scientific data.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

This paper presents a neural scheme for the topology-aware interpolation of time-varying scalar fields. Given a time-varying sequence of persistence diagrams, along with a sparse temporal sampling of the corresponding scalar fields, denoted as keyframes, our interpolation approach aims at "inverting" the non-keyframe diagrams to produce plausible estimations of the corresponding, missing data. For this, we rely on a neural architecture which learns the relation from a time value to the corresponding scalar field, based on the keyframe examples, and reliably extends this relation to the non-keyframe time steps. We show how augmenting this architecture with specific topological losses exploiting the input diagrams both improves the geometrical and topological reconstruction of the non-keyframe time steps. At query time, given an input time value for which an interpolation is desired, our approach instantaneously produces an output, via a single propagation of the time input through the network. Experiments interpolating 2D and 3D time-varying datasets show our approach superiority, both in terms of data and topological fitting, with regard to reference interpolation schemes.

Problem

• Interpolate missing scalar fields from sparse temporal keyframes
• Invert non-keyframe persistence diagrams to estimate missing data
• Improve geometric and topological reconstruction using topological losses

Innovation

• Neural architecture learning time-to-field mapping
• Topological losses enhance geometric reconstruction
• Instant interpolation via single network propagation